Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9369 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#386 |
| prime radical | #60790 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1245 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,9,6,8,4,14)(3,10,13,15,12,16,11,7)(17,18)(19,23,29,32)(20,25)(21,27)(22,26)(24,28,31,30);(1,3)(2,7,8,15)(4,11)(5,13)(6,12)(9,10,14,16)(17,19)(18,23,26,32)(20,28)(21,30)(22,29)(24,25,31,27);(2,8)(4,5)(7,15)(11,13)(17,19)(18,24)(20,28)(21,30)(22,29)(23,25)(26,31)(27,32);(17,20,22,21)(18,25,26,27)(19,28,29,30)(23,31,32,24)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=i,e^2=hi,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,e^b=ei,f^b=fh,g^b=gi,e^c=eh,f^c=fi,g^c=gh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 107518);